The costate equation is related to the state equation used in
optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
of first order
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s
:
where the right-hand side is the vector of
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s of the negative of the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
with respect to the state variables.
Interpretation
The costate variables
can be interpreted as
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
associated with the state equations. The state equations represent constraints of the minimization problem, and the costate variables represent the
marginal cost
In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it r ...
of violating those constraints; in economic terms the costate variables are the
shadow price
A shadow price is the monetary value assigned to an abstract or intangible commodity which is not traded in the marketplace. This often takes the form of an externality. Shadow prices are also known as the recalculation of known market prices in o ...
s.
Solution
The state equation is subject to an initial condition and is solved forwards in time. The costate equation must satisfy a
transversality condition In optimal control theory, a transversality condition is a Boundary value problem, boundary condition for the terminal values of the costate equation, costate variables. They are one of the necessary conditions for optimality infinite-horizon optima ...
and is solved backwards in time, from the final time towards the beginning. For more details see
Pontryagin's maximum principle
Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it ...
.
[ Ross, I. M. ''A Primer on Pontryagin's Principle in Optimal Control'', Collegiate Publishers, 2009. .]
See also
*
Adjoint equation An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equat ...
*
Covector mapping principle
The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by Ross and co-workers,Ross, I. M., “A Historical Introduction to the Covector Mappin ...
*
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
References
{{DEFAULTSORT:Costate Equation
Optimal control
Calculus of variations